Gauss's constant

In mathematics, Gauss's constant, denoted by G, is defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2:

The constant is named after Carl Friedrich Gauss, who in 1799[1] discovered that

so that

where Β denotes the beta function.

Relations to other constantsEdit

Gauss's constant may be used to express the gamma function at argument 1/4:

 

Alternatively,

 

and since π and Γ(1/4) are algebraically independent, Gauss's constant is transcendental.

Lemniscate constantsEdit

Gauss's constant may be used in the definition of the lemniscate constants, the first of which is:

 

and the second constant:

 

which arise in finding the arc length of a lemniscate. Both constants were proven to be transcendental.[2]

Other formulasEdit

A formula for G in terms of Jacobi theta functions is given by

 

as well as the rapidly converging series

 

The constant is also given by the infinite product

 

It appears in the evaluation of the integrals

 
 

Gauss' constant as a continued fraction is [0, 1, 5, 21, 3, 4, 14, ...]. (sequence A053002 in the OEIS)

See alsoEdit

ReferencesEdit

  1. ^ Nielsen, Mikkel Slot. (July 2016). Undergraduate convexity : problems and solutions. p. 162. ISBN 9789813146211. OCLC 951172848.
  2. ^ Todd, John (1975). "The lemniscate constants". ACM DL.